Nick Powell struck the bar for Sir Alex Ferguson's side, which was finally beaten in the Champions League this season by a Turkish side that kept its knockout round hopes alive.
Burak Yilmaz scored the only goal of the game as he headed powerfully beyond Anders Lindegaard, who made several good saves to deny the Turkish side.
Nick Powell hit the bar on his first competitive start for United, but the home side was in control for the majority of the game and kept its hopes of progression to the knockout stages alive.
With qualification for the knockout stages already wrapped up, Sir Alex Ferguson rested a number of big names and welcomed back Phil Jones for his first game of the season after injury.
The visiting side had a chance in the second minute as Welbeck latched onto a poor Nordin Amrabat pass and fed Tom Cleverley, but the England international’s shot deflected into Fernando Muslera’s arms.
Burak was quick to respond for the home side with a long-range effort which Lindegaard fumbled and Jones cleared to safety.
The Turkey international could have opened the scoring two more times inside the first 25 minutes, but first failed to connect with a corner after some poor defending, before heading a free kick over the bar.
Galatasaray kept the pressure on and tested Lindegaard once more, but the 28-year-old firmly pushed away a stinging drive from Hamit Altintop.
The home side felt it should have had a penalty shortly after, when a Selcuk Inan cross struck Alexander Buttner’s arm, but appeals were turned down by referee Carlos Velasco.
Despite the home team's dominance the Old Trafford club had the best chance of the half from a Tom Cleverley corner which was met at the back post by Powell and beat Muslera but rattled the bar and bounced clear.
Galatasaray remained on top at the start of the second half and Lindegaard was again called into action, this time diving to his right and holding onto a sweetly-struck Albert Riera volley.
Lindegaard made another save from close range to keep out a Felipe Melo header from an Emmanuel Eboue cross, but there was little the Dane could do about the opening goal as Burak rose to meet a Selcuk corner and power it into the net.
United stirred after falling behind and nearly equalized on the counterattack, but Javier Hernandez dallied over his shot and saw it deflected safely into Muslera’s grasp.
With 15 minutes of normal time to play Sir Alex Ferguson made a double substitution, replacing Powell and Anderson with Ashley Young and Federico Macheda.
Yet, it was Galatasaray which would go close again through Altintop, whose thunderous strike from 20 yards was only kept out by the bar after a fingertip save from Lindegaard.
Sir Alex Ferguson made one final change, handing a Champions League debut to Josh King, who replaced Danny Welbeck.
A weak Hernandez strike which was comfortably saved by Muslera would be the last chance of the game for the group leader, which suffered its first defeat in the competition this season.
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